Maps preserving some spectral domains of skew products of operators

Abstract

Let be an infinite dimensional Hilbert space and be the algebra of all bounded linear operators on . For , let denote the semi-Fredholm domain, the Fredholm domain or the Weyl domain of A in the spectrum, σ(A). We characterize the form of a map ϕ from into itself with range contains the operators of rank at most two and satisfies for all . We also describe those maps on preserving the same spectral set ‘but of Jordan skew-triple product’ ‘of operators’.